User blog:Cheetahrock63/Abstract polytopes
Abstract polytopes are by far the broadest notion of a polytope—when considering them, the only things one wants to consider are how many of each subfacets (commonly known as a “face” or “element” or “ k -face(t)” where k is the dimension of the face) there are and how the faces relate to one another. No internal angles, no edge lengths, no circumradii or inradii, no need for the edges to be straight lines, no skewness, no need to consider the curvature of space the polytope needs to be embedded in or how edges intersect each other, none of that. A way to think about it is that the only parts that abstract polytopists directly see in tetrahedra are that there are 4 vertices, 6 edges, 4 triangular faces, 1 tetrahedral cell, and the relationships between the subfacets with regards to which subfacets contain what other subfacets. The relationships between the subfacets can be tabulated with thingies known as incidence matrices. Basically two subfacets are “incident” if one contains the other or if they’re the same subfacet. The incident matrix is a table with all of the subfacets laid out in the rows and columns. The entry in the row for subfacet x and the column for subfacet y is 1 if x and y are incident and 0 if they are not. Incidence matrix of a tetrahedron ABCD Literally the simplest (non-dihedral, non-hosohedral) polyhedron and it has an incidence matrix that looks like that. So small and pathetic. 0/10. Needs more insanity. In all seriousness though, incidence matrices are in general, really, really big and hard to make in the first place. A much smaller and compact method of tabulating incidences are configuration matrices. Below is the configuration matrix of a tetrahedron. Instead of a 16x16 matrix for 16 subfacets of a tetrahedron, the configuration matrix of the tetrahedron is know a 5x5 matrix for 5 types of subfacets of a tetrahedron. Likewise, instead of a 10442x10442 for 10442 subfacets of a mom fapathi, the configuration matrix is now a 11x11 matrix for 11 types of subfacets. Also, “tilings” and “honeycombs” are now considered just special cases of polytopes. When one does need to consider other properties, they have to use realizations of the polytopes. The most commonly used space to realize polytopes is course, flat Euclidean space. Of course, since one’s not concerned about silly stuff like whether or not edges of polytopes have to intersect one another or whether or not you have to stuff the polytope in a hyperbolic space and is only concerned with how many subfacets there are, we don’t need to put crazy complex numbers inside Schläfli symbols and/or consider Schläflians or anything like that. No, regular polytopes and their truncated varieties need cardinal numbers in their symbols. Apeirogons, for example, have a symbol of \{\aleph_0\} . Star polytopes, failed star polytopes, pseudogons, and synkrotopes are just special realizations of some kind of polytope. Quick overview of the symbols of some complexgons: A star polygon with symbol \{\frac{p}{q}\} where p and q are coprime integers greater than 0 now has a symbol of just \{p\} . This makes sense since a \frac{p}{q} -gon has the same number of vertices and edges as a p , but it’s only the way the edges connect the vertices that may differ—a factor we now shouldn’t care about. Any failed star polygon (which has a symbol of \{x\} where x is an irrational real number), or pseudogon (a symbol of \{z\} where \{z\} is a nonzero pure imaginary), or synkrogon (a symbol of \{z\} where \{z\} is any complex number that doesn’t square to a real number) now just has a symbol of \{\aleph_0\} . This makes sense since such polygons do have \aleph_0 vertices and edges and it wouldn’t hurt at all to think of them as just different kinds of apeirogon (failed star polygons are “circular apeirogons”, regular apeirogons are “Euclidean apeirogons”, pseudogons are “hyperbolic apeirogons”, and synkrogons such as goldengons are “synkrobolic apeirogons”). In general, any entries p: p \in \mathbb{C} \backslash \{0\} of a Schläfli symbol of traditional polytopes should now become the cardinal {\rm {card}}(\{\exp (\frac{i\tau n}{p})|n \in \mathbb {Z}\}) . We say that as abstract polytopes, \{p\} is isomorphic to \ (\{\exp (\frac{i\tau n}{p})|n \in \mathbb {Z}\})\} . So disregarding anything but combinatorials, \{\frac{12}{5}\} (dodecagram \{12i\} (12i-gon), \{3, \frac{5}{2}\} (great icosahedron), \{\frac{5}{2}, 4\} (order-4 pentagrammic tiling), and \{\frac{7}{3}, \infty, \sqrt{3}, 12i, \frac{9}{4}, 69+420i\} (order-(√3)-(12i)-(9/4)-(69+420i) infinite order great heptagrammic hexacomb) will now be intepreted as \{12\} (dodecagon), \{\aleph_0\} (apeirogon), \{3, 5\} (icosahedron), \{5,4\} (order-4 pentagonal tiling), and \{7,\aleph_0, \aleph_0, \aleph_0, 9, \aleph_0\} (order- \aleph_0 - \aleph_0 —9- \aleph_0 order- \aleph_0 heptagonal hexacomb) respectively. Yes, it is true that, say, the order-4 pentagrammic tiling is a spherical tiling (a failed star polyhedron) and the order-4 pentagonal tiling is traditionally realized as a hyperbolic planar tiling but technically, pentagrams are pentagons since they both have five edges (the numbers of subfacets are only thing that we care about so that means pentagrams and pentagons are exactly the same thing) so technically, an order-4 pentagrammic tiling is an order-4 pentagonal tiling. Plus, we have to ignore the curvature of the space realizations have to be embedded in when we treat them as abstract polytopes. Slightly more formally, an abstract polytope P is a partially ordered set (poset) with the partial order \leq ( X: X \in P is incident to Y: Y \in P iff X \leq Y \or Y \leq X ) and the corresponding strict partial order < ( F < G \iff ((F \leq G) \neg \land (F = G)) ) whose elements are called “faces” or “subfacets”, whose totally ordered subsets of the greatest possible cardinality are called “'flags'”, and whose closed intervals are called “'sections'” such that: *There exists a least face and a greatest face in P . The least face of any polytope is the null polytope (denoted \emptyset or with Schläfli symbol )( ) and the greatest face of any polytope is the “interior of the polytope”. Faces of P that are neither the greatest nor the least are called proper faces of P . *The cardinalities of all flags of P are all equal. (specifically, this cardinality is the rank plus 2 of P ) * P is strongly connected. A poset Q is connected if for any two proper faces F , G , of Q , there exists a sequence of faces H_n: n \in \mathbb {N} \land 1 \leq n \leq k such that H_1 = F , H_k = G , and for all n \in \mathbb {N}: n < k , H_n is incident with H_{n+1} . A poset is strongly connected if all of its sections are connected. *If the ranks of two faces F and H differ by 2, there exist only two faces H such that F < G < H . This is known as the “diamond property”. The rank or dimensionality of a polytope is the rank of its greatest face, the rank plus two of any non-null polytopic face F of is the cardinality of a chain from \emptyset to F . Since the rank plus two of the null polytope is 1, the rank of a null polytope ends up being -1. This is where the idea of the null polytope being “negative first dimensional” comes from. The only polytope of rank 0 is the point, faces of rank 0 are called vertices, the only polytope of rank 1 is the line segment, faces of rank 1 are called edges, polytopes of rank 2 are called polygons, faces of rank 2 are called faces, polytopes of rank 3 are called polyhedra, faces of rank 3 are called cells, polytopes of rank 4 are called polychora, faces of rank 4 are called tera, and so on. Faces of rank k-1 are called facets, faces of rank k-2 are called ridges, faces of rank k-3 are called peaks, faces of rank k-4 are called spires. Traditionally, polytopes are formally defined in a way that makes F < G equivalent to F \in G (which means calling subfacets “elements” is really not a stretch). A vertex of a polytope V_n (where n is an index, generally a natural/integer/surinteger) can be defined as so: V_n = \{\emptyset, \{n\}\} (that way, for any given point, \emptyset is an element of V_n ). An edge with vertices V_n V_m is be defined as \{\emptyset, V_n, V_m\} ; a face can be defined as the set containing the empty set, all of its vertices, and all of its edges; and so on. So we can now define a tetrahedron ABCD as the poset T (whose partial order is the element-of relation) so: * T := \{\emptyset, F\} \cup \{V_n|n \in \mathbb {N} \land 0 \leq n \leq 3\} \cup \{E_n|n \in \mathbb {N} \land 0 \leq n \leq 5\} * A := V_0 , B := V_1 , C := V_2 , D := V_3 * E_0 = AB := \{\emptyset, A, B\} , E_1 = AC := \{\emptyset, A, C\} , E_2 = AD := \{\emptyset, A, D\} , E_3 = BC := \{\emptyset, B, C\} , E_4 = BD := \{\emptyset, B, D\} , E_5 = CD := \{\emptyset, C, D\} * F := \{\emptyset\} \cup \{V_n|n \in \mathbb {N} \land 0 \leq n \leq 3\} \cup \{E_n|n \in \mathbb {N} \land 0 \leq n \leq 5\} Anyway, the definition of an abstract polytope actually prevents monogons or (hyper)complex polytopes in general from being polytopes (they do not satisfy the diamond property), but also allows for a shit-ton of objects never before seen in spherical, Euclidean, or hyperbolic space such as toroidal polytopes (e.g. Császár polyhedra and Szilassi polyhedra) or real projective polytopes or Klein bottle polytopes or polytopes that literally cannot be tilings of any manifold. Examples of real projective polytopes include the hemi-cube, hemi-octahedron, hemi-dodecahedron, and the hemi-icosahedron. All of which have the same types of faces and verfs (the verf of a polytope P with greatest face P’ at a vertex p is the section P’ ) as their non-hemi spherical counterparts. This implies that Schläfli symbols—symbols literally defined from the faces and verfs of a polytope—are no longer unique to a single polytope. Polytopes can share symbols now. And it’s not just true for the hemis, for example, the hendecachoron and pentacontaheptachoron—polychora that literally cannot be realized as tilings of any manifold (for such a space wouldn’t be locally Euclidean but “locally projective”)—share symbols with the icosahedral honeycomb ( \{3,5,3\} ) and order-5 dodecahedral honeycomb ( \{5,3,5\} ) respectively. The dodecadodecahedron ( t_1 \{\frac{5}{2}, 5\} )—a uniform polytope realized with pentagonal and pentagrammic faces—can now be seen as a toroidal order-4 pentagonal tiling and a now a regular polyhedron with symbol \{5,4\} —a symbol shared with the hyperbolic order-4 pentagonal tiling. (Keep in mind that the pentagrams in the dodecadodecahedron are treated as pentagons.) The dodecadodecahedron is an example of a non-regular polyhedron not isomorphic to any regular traditional polyhedron that is combinatorially regular as an abstract polytope but not regular as a traditional polytopehttp://homepages.wmich.edu/~drichter/regularpolyhedra.htm, examples of objects that I’ll call pseudoregular polytopes. Further specification is sometimes used within the symbols of polytopes to compensate for that (e.g. the symbol for hemi-dodecahedron being \{5,3\}_5 or \frac{\{5,3\}}{2} . The former specification is known as a McMullen symbol while the latter is known as a Coxeter symbol. McMullen symbols will be used here on out.). An abstract polytope is considered “locally X ” (where X is some manifold) if the manifolds the polytope’s facets and verfs can be realized as tilings of can be topologically either hyperspherical/Euclidean or X . Locally spherical polytopes are tilings of manifolds and include the familiar faces. When X is non-spherical, the polytope is not a tiling of any manifold in the usual sense. Other polytopes that aren’t tilings of manifolds include polytopes whose faces and verfs both cannot tile anything topologically a hypersphere or Euclidean space. One interesting thing about abstract polytopes is that a natural generalization for infinite cardinals larger than \aleph_0 to reasonably be entries in Schläfli symbols can be found very easily. For example, one can consider \{\aleph_1\} —an example of a long apeirogon. Long apeirogons can exist as tilings of a long line—a line that’s basically made up of uncountably many half-open line segments glued end-to-end (the most famous example is the standard order topology on (\omega_1^* \times (0,1)) \cup (\omega_1 \times [0,1)) . Here, instead of the order type of an ordinal like \omega_1 , the order type of a set of surintegers (“omnific integers”) with birthday less than the initial ordinal of some infinite cardinal is used.). Longer than a long line would be something an \{\aleph_2\} would tile—it’s what should clearly and obviously be called the “'longer line'”. So, \{\aleph_2\} will be called a “'longer apeirogon'”. \{\aleph_3\} should then very definitely be called the longerer apeirogon or long-3-er apeirogon. In general, the analogue of a long line made from \aleph_\alpha half-open line segments should totally be a long-α-er line and a polygon with symbol \{\aleph_\alpha\} will be a long-α-er apeirogon. Now unfortunately, longer lines or anything longer than that are not locally homeomorphic to the Euclidean line so they are not manifolds and the long-ass apeirogons and whatnot aren’t actually tilings of manifolds. So sad. But being not tilings of any manifolds hasn’t exactly stopped many other abstract polytopes, amirite? Unfortunately, what does stop them is their inability to satisfy strong connectedness. All polygons with more than \aleph_0 edges do not satisfy that property and are therefore not actually abstract polytopes and are actually isomorphic to compounds of \{\aleph_0\} . \{\kappa\} for cardinal greater than \aleph_0 is isomorphic to a compound of \kappa apeirogons. So sad. A list of all regular abstract polytopes with dimensionality greater than or equal to 1 but less than or equal to 9 that have a flag count less than or equal to 2000 but not equal to 1024 or 1536 is available here. Note how many polytopes share the same Schläfli symbol. Proofs Monogons are not abstract polytopes A monogon A = \{\emptyset, V_0, E_0, F_0\} has one vertex V_0 , one edge E_0 , and is one face F_0 . By definition of an abstract polytope, there should be two subfacets between any two subfacets that differ in rank by 2 (diamond property). In the case of the monogon, any two subfacets that differ in rank by 2 have only one subfacet in between (which means that monogons are totally ordered). Because the diamond property isn't satisfied, There are complex polytopes are not abstract polytopes Proof by counterexample. A tricomtelon is an example of a complex polytelon. It has three vertices and is one edge. By definition of an abstract polytope, there should be two subfacets between any two subfacets that differ in rank by 2 (diamond property). The edge of the tricomtelon and the null polytope differ in rank by two and there are three subfacets in between them (that is, the three vertices). Therefore, tricomtela and any complex polytopes that have them as edges do not satisfy the diamond property and are not abstract polytopes Definition for polygons with more than \aleph_0 edges; such polytopes are not abstract polytopes An infinite-edged polygon (P, <, \leq) can be defined as so: * P = \{\emptyset, F\} \cup \{E_n|n \in O\} \cup \{V_n|n \in O\} . O is the set of all surintegers with birthday less than an arbitrary initial ordinal of an infinite cardinal \kappa . The number of edges is greater than \aleph_0 iff \kappa is more than \aleph_0 . * \forall f \in P (\emptyset \leq f) * \forall f \in P (f \leq F) * \forall f \in P \forall n \in O(V_n < f \implies (f = F \or f = E_n \or E_{n+1})) ---- * V_n := \{\emptyset, \{n\}\} * E_n := \{\emptyset, V_n, V_{n+1}\} * F := \{\emptyset\} \cup \{E_n|n \in O\} \cup \{V_n|n \in O\} * P := F \cup \{F\} * F < G \iff F \in G * F \leq G \implies F \in G \or F = G P has a least and greatest face ( \emptyset and F respectively), all flags of P (which are either in the form \{\emptyset, V_n, E_n, F\} or \{\emptyset, V_n, E_{n+1}, F\} ) have a cardinality of 4, and the diamond property is satisfied. However for polygons with more than \aleph_0 edges, the condition that an abstract polytope should be strongly connected is not satisfied so they are therefore not abstract polytopes. Pseudoregular polytopes A non-regular traditional polytope is called pseudoregular if as an abstract polytope, it is regular and not isomorphic to any regular traditional polytopes as an abstract polytope. Pseudoregular polytopes can be characterized by their facets and verfs: any given facet of a pseudoregular is combinatorially regular and isomorphic to all other facets of the polytope as abstract polytopes. Any given verf of a pseudoregular is combinatorially regular and isomorphic to all other verfs of the polytope. Five examples of pseudoregular polyhedra (aka “'regular polyhedra of index two'”) are: *the dodecadodecahedron (did), Schläfli symbol t_1 \{\frac{5}{2}, 5\} (traditional), \{5, 4\}_6 (abstract); has 12 pentagonal faces and 12 pentagrammic faces (which are isomorphic to pentagons); has 24 rectangular verfs (which are isomorphic to squares) *the medial rhombic triacontahedron; the dual of did; Schläfli symbol \{4,5\}_6 (abstract); has 32 rhombic faces (isomorphic to squares); has 12 pentagonal verfs and 12 pentagrammic verfs (isomorphic to pentagons) *the ditrigonal dodecadodecahedron (ditdid), Schläfli symbol b \{5, \frac{5}{2}\} (traditional), \{5, 6\}_4 (abstract); has 12 pentagonal faces and 12 pentagrammic faces (isomorphic to pentagons); has 24 propeller tripodal verfs (isomorphic to hexagons) *the medial triambic icosahedron (Df1g1); dual of ditdid; Schläfli symbol \{6,5\}_4 (abstract); has 20 simple concave isogonal hexagonal faces (isomorphic to hexagons); has 12 pentagonal verfs and 12 pentagrammic verfs (isomorphic to pentagons) *the excavated dodecahedron (Ef1g1); Schläfli symbol \{6,6\}_6 (abstract); has 20 propeller tripodal faces (isomorphic to hexagons); has 20 simple concave isogonal hexagonal verfs (isomorphic to hexagons) References Category:Blog posts